Quadratic Equations and Imaginary Numbers Learning Targets

Front

What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What does it mean to rewrite a quadratic equation as a perfect square?

Back

Rewriting a quadratic equation as a perfect square means expressing it in the form (x - p)² = q, where p and q are constants.

Front

How do you complete the square for a quadratic equation?

Back

To complete the square, rearrange the equation to isolate the constant term, then add and subtract the square of half the coefficient of x.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac, which determines the nature of the roots.

Front

What do the values of the discriminant indicate?

Back

If the discriminant is positive, there are two distinct real solutions; if zero, there is one real solution; if negative, there are two complex solutions.

Front

What are imaginary numbers?

Back

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit i, where i = √(-1).

Front

How do you simplify the square root of a negative number?

Back

To simplify the square root of a negative number, factor out -1 and use the property √(-1) = i.

Front

What are conjugate pairs in the context of quadratic equations?

Back

Conjugate pairs are two complex solutions of a quadratic equation that are of the form a + bi and a - bi, where a and b are real numbers.

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of a quadratic equation.

Front

How can you determine if a quadratic equation has real solutions?

Back

You can determine if a quadratic equation has real solutions by evaluating the discriminant; if it is non-negative, real solutions exist.